Optimal destabilization of K-unstable Fano varieties via stability thresholds

Abstract

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Székelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.

arXiv commentFinal version, to appear in Geom. Topol

References in corpus (30)

• A variational approach to the Yau-Tian-Donaldson conjecture
• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• K-semistability is equivariant volume minimization
• Optimal destabilizing centers and equivariant K-stability
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below
• Existence of valuations with smallest normalized volume
• Greatest lower bounds on the Ricci curvature of Fano manifolds
• Reductivity of the automorphism group of K-polystable Fano varieties
• A non-Archimedean approach to K-stability
• Algebraicity of the Metric Tangent Cones and Equivariant K-stability
• Kähler-Ricci flow, Kähler-Einstein metric, and K-stability
• On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
• Stability of Valuations and Kollár Components
• The volume of singular Kähler-Einstein Fano varieties
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• The normalized volume of a singularity is lower semicontinuous
• K-stability of log Fano hyperplane arrangements
• Uniqueness of K-polystable degenerations of Fano varieties
• Wall crossing for K-moduli spaces of plane curves
• The existence of the Kähler-Ricci soliton degeneration
• Uniqueness of the minimizer of the normalized volume function
• Finite generation for valuations computing stability thresholds and applications to K-stability
• Openness of K-semistability for Fano varieties
• Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below
• Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold
• Fano varieties with large Seshadri constants
• On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows
• Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

Cited by in corpus (9)

• K-stability of Fano varieties: an algebro-geometric approach
• On properness of K-moduli spaces and optimal degenerations of Fano varieties
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Equivariant K-stability under finite group action
• Toward finite generation of higher rational rank valuations
• On wall crossing for K-stability
• On K-stability of Calabi-Yau fibrations
• Log K-stability of GIT-stable divisors on Fano varieties
• Approximating delta invariants in the sense of complements of plt type